where \(t_n\)  is time of nucleation of a bubble in the Hubble volume, and \(t_{\rm PBH}\)  is the time of production of the PBH.
The production time \(t_{\rm PBH}\)  can be obtained when the density contrast between the inside and outside of the Hubble volume exceeds the critical value \(\delta _c=0.45\)  [1]}, [2]}.
The energy density contrast \(\delta \)  is given by Eq. (REF ).
When \(\delta \)  exceeds \(\delta _c\) ,
the inside of the Hubble volume can gravitationally collapse into a PBH.
The PBH mass is given by
\(M_{\rm PBH} \sim \frac{4\pi }{3}H^{-3} (t_{\rm PBH}) \rho _c = 4\pi H^{-1} (t_{\rm PBH}).\) 
